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Behavior of the Torsion of the Differential Module of an Algebroid Curve Under Quadratic Transformations

In: Algebra, Arithmetic and Geometry with Applications

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  • Robert W. Berger

Abstract

The question, whether the torsion submodule Τ of the differential module of the local ring R of a singular point of an algebraic or algebroid curve is not zero, is still open in general. All examples suggest the conjecture that this torsion genuinely decreases when going from R to the first quadratic transform R 1, which would imply that Τ was nontrivial in the first place. We give a general formula for the difference $$ \ell _R (T) - \ell _{R_1 } (T_1 )$$ of the lengths of these torsions. In the special cases that R is a semigroup ring which is a complete intersection or that R is a “nice” almost complete intersection or a “stable” complete intersection (definition 1) the conjecture is proved.

Suggested Citation

  • Robert W. Berger, 2004. "Behavior of the Torsion of the Differential Module of an Algebroid Curve Under Quadratic Transformations," Springer Books, in: Chris Christensen & Avinash Sathaye & Ganesh Sundaram & Chandrajit Bajaj (ed.), Algebra, Arithmetic and Geometry with Applications, pages 189-201, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-18487-1_10
    DOI: 10.1007/978-3-642-18487-1_10
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